The simple Brownian bug model with no interaction, i.e., when
the birth and death rates of the individuals are given by
Eq. (REF ), has been studied and discussed in various
works [1]}, [2]}. The ensemble average of the
total population size follows
\( \langle N(t) \rangle = N_0 \exp {[\Delta (t - t_0)]} \, ,\) 
